Small deformations of polygons and polyhedra

Schlenker, Jean-Marc

doi:10.1090/S0002-9947-06-04172-9

Small deformations of polygons and polyhedra
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Trans. Amer. Math. Soc. 359 (2007), 2155-2189 Request permission

Abstract:

We describe the first-order variations of the angles of Euclidean, spherical or hyperbolic polygons under infinitesimal deformations such that the lengths of the edges do not change. Using this description, we introduce a vector-valued quadratic invariant $b$ on the space of those isometric deformations which, for convex polygons, has a remarkable positivity property. We give two geometric applications. The first is an isoperimetric statement for hyperbolic polygons: Among the convex hyperbolic polygons with given edge lengths, there is a unique polygon with vertices on a circle, a horocycle, or on one connected component of the space of points at constant distance from a geodesic, and it has maximal area. The second application is a rigidity result for equivariant polyhedral surfaces in the Minkowski space. Résumé. On décrit les déformations infinitésimales des angles d’un polygone euclidien, sphérique ou hyperbolique sous les déformations infinitésimales qui préservent les longueurs des arêtes. On en déduit la définition d’un invariant quadratique à valeurs vectorielles $b$ sur l’espace de ces déformations isométriques qui, pour les polygones convexes, a une propriété remarquable de positivité. On donne deux applications géométriques. La première est un énoncé isoperimétrique pour les polygones hyperboliques : Parmi les polygones hyperboliques convexes dont les longueurs des arêtes sont données, il existe un unique élément dont les sommets sont sur un cercle, un horocycle, ou dans une composante connexe de l’ensemble des points à distance constante d’une géodésique, et son aire est maximale. La seconde application est un résultat de rigidité pour les surfaces polyèdrales équivariantes dans l’espace de Minkowski.

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